Multi-lumen cannula

ABSTRACT

This document relates to methods and materials for providing blood flow for a blood pump recipient. For example, cannulae that can be connected to the circulatory system of a mammal and can be used in conjunction with a blood pump (e.g., an assist device) are provided.

BACKGROUND

1. Technical Field

This document relates to methods and materials for providing blood flowfor a blood pump recipient. For example, this document provides cannulaethat can be connected to the circulatory system of a mammal and can beused in conjunction with a blood pump (e.g., an assist device).

2. Background Information

Mechanical circulatory support (MCS) is a way of improving blood flow ina failing heart using an electrically or pneumatically powered bloodpump. A ventricular assist device (VAD) is an implantable blood pumpthat works in conjunction with the recipient's own heart to pumpsufficient blood throughout the body. Heart failure may affect the rightside of the heart, limiting the ability of the heart to pump blood tothe lungs, or the left side of the heart, resulting in an inability topump sufficient oxygen-rich blood to the rest of the body. Often, bothsides of the heart are affected. A VAD can provide short-term MCS whilea recipient is awaiting cardiac transplant, or permanent MCS for arecipient who is not a candidate for transplantation, by deliveringconsistent blood flow to vital organs.

SUMMARY

This document relates to methods and materials for providing blood flowfor a blood pump recipient. For example, this document provides cannulaethat can be connected to the circulatory system of a mammal and can beused in conjunction with a blood pump (e.g., an assist device). In somecases, a cannula provided herein can have an eccentric multi-lumendesign that can require a single insertion site. For example, a cannula,featuring a circular-shaped lumen nested in the notch of areniform-shaped lumen and a thin, flexible septum, is provided. Cannulaeprovided herein can provide a blood-flow path with beneficial fluiddynamics and can reduce the complexity associated with blood pumpplacement.

In general, a cannula for use with a blood pump is described. Thecannula includes a housing having a proximal region, a distal region,and an intermediate region located between the proximal and distalregions. The housing defines a first lumen and a second lumen. The firstlumen includes (a) a proximal end located within the proximal region ofthe housing and adapted to engage the blood pump and (b) a distal endlocated within the distal region of the housing and adapted to bepositioned within a cardiovascular system. The second lumen comprises(a) a proximal end located within the proximal region of the housing andadapted to engage the blood pump and (b) a distal end located within theintermediate region of the housing and adapted to be positioned withinthe cardiovascular system. One of the first and second lumens has agenerally reniform cross-sectional shape in the intermediate region.

In another aspect, this document describes a method for implanting acannula as described above into the heart of a mammal. The methodcomprises, or consists essentially of, puncturing the heart or a bloodvessel of the mammal, and inserting a cannula into the chamber of theheart, so that the distal region of the cannula is positioned within ablood vessel of the mammal and the intermediate region of the cannula ispositioned within a chamber of the heart of the mammal. The method caninclude connecting a blood pump to the proximal region of the cannula.The blood pump can receive blood from the heart through the second lumenof the cannula and pump blood to the blood vessel through the firstlumen of the cannula. The distal end of the first lumen can bepositioned in the aorta and the distal end of the second lumen can bepositioned in the left ventricle. In another aspect, the distal end ofthe first lumen can be positioned in the pulmonary artery and the distalend of the second lumen can be positioned in the right ventricle.

In another aspect, this document describes a system for providing bloodflow to a mammal. The system comprises, or consists essentially, of acannula as described above and a blood pump. The proximal end of thefirst lumen of the cannula can be connected to the inflow of the bloodpump and the proximal end of the second lumen of the cannula can beconnected to the outflow of the blood pump.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention pertains. Although methods and materialssimilar or equivalent to those described herein can be used to practicethe invention, suitable methods and materials are described below. Allpublications, patent applications, patents, and other referencesmentioned herein are incorporated by reference in their entirety. Incase of conflict, the present specification, including definitions, willcontrol. In addition, the materials, methods, and examples areillustrative only and not intended to be limiting.

The details of one or more embodiments of the invention are set forth inthe accompanying drawings and the description below. Other features,objects, and advantages of the invention will be apparent from thedescription and drawings, and from the claims.

DESCRIPTION OF DRAWINGS

FIG. 1 is a perspective view of the housing of one exemplary embodimentof a cannula.

FIG. 2 is a cross-sectional view of the intermediate region of thehousing as depicted in FIG. 1.

FIG. 3 is front view of a cannula connected to a blood pump.

FIG. 4 is a schematic depicting geometric relationships of lumens.

FIG. 5 is a representation of an implanted pump with a cannula insertedin the right ventricle of a person.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

This document relates to methods and materials for providing blood flowfor a blood pump recipient. For example, this document provides cannulaethat can be connected to the circulatory system of a mammal and can beused in conjunction with a blood pump (e.g., an assist device). In somecases, a cannula provided herein can have an eccentric multi-lumendesign that can require a single insertion site. For example, a cannulacan feature a circular-shaped lumen nested in the notch of areniform-shaped lumen and can feature a thin, flexible septum.

In reference to FIG. 1, cannula 11 for use with a blood pump can beconstructed from single housing 10 having proximal region 18, distalregion 32, and intermediate region 26 located between region 18 andregion 32. Housing 10 can define first lumen 14 and second lumen 16.First lumen 14 can have proximal end 24 located within proximal region18 of housing 10 and distal end 34 located within distal region 32 ofhousing 10. Second lumen 16 can have proximal end 25 located withinproximal region 18 and distal end 35 located within intermediate region26.

Proximal ends 24 and 25 can be adapted to engage a blood pumpreleasably. As used herein, “adapted to releasably engage a blood pump”refers to any feature that allows interchangeability of a blood pumpwithout the need to remove cannula 11 from the circulatory system. Forexample, proximal ends 24 and 25 can be constructed to permit connectionto a blood pump via additional tubing, rigid fittings, or connectors. Insome cases, proximal ends 24 and 25 can include additional tubing of anylength or diameter suitable for being connected to a blood pump. In somecases, additional tubing can have multiple segments, which may be areasin which additional tubing is compressed and folded back on itself, topermit for articulation of additional tubing. For example, theconstruction of the tubing may be similar to the construction of aflexible drinking straw, but providing an inner surface that is morerounded than that of a drinking straw. In other examples, the tubing maybe constructed so that the tubing segments are similar to crimps in avascular graph, convolutions in a pair of bellows, or fluting withincorrugated cardboard. In some cases, additional tubing may be flared atboth ends to engage proximal end 24 or 25 at one end and a blood pump atanother end. In some cases, additional tubing may be bonded orcompressed onto a separate rigid fitting for connection to a blood pump.

In some cases, releasable engagement of proximal ends 24 and 25 can beaccomplished with connectors of an appropriate form that allow forreliable connections. For example, appropriate connectors can be screwrings that can be tightened to form a fluid-tight seal so as to preventfluids from leaking out of the system during operation. (See e.g., U.S.Pat. Pub. No. 2006/0074271). Other suitable connectors can includetwist-and-lock connectors, connectors with bolted flanges,circumferential clamps, compressive fitting, snap fit, or simplethreaded connections. Suitable connectors can be provided withappropriate locking features that prevent them from loosening after ablood pump has been implanted. Various connectors can be sized to allowinterchangeability of a blood pump without the need to remove thecannula from the circulatory system. In some cases, proximal ends 24 and25 can be adapted to engage a blood pump by different engagement means.For example, if proximal end 24 is adapted to engage the outflow port ofa blood pump, proximal end 25 can be adapted to engage the inflow portof the blood pump.

Additionally, in some implementations, the cannula or cannula structuremay be permanently attached to the pump. For example, the inflow oroutflow portions of the cannula could be integrated into the pumphousing.

Distal ends 34 and 35 can be adapted to be positioned within acardiovascular system. In some cases, distal ends 34 and 35 canbifurcate or branch from the intermediate region of the cannula. As usedherein, “adapted to be positioned within a cardiovascular system” refersto any feature that permits cannula 11 to be inserted into a heart orvasculature of a blood pump recipient and provides for appropriate bloodflow through lumens 14 and 16. In some cases, distal ends 34 and 35 canhave a single opening in line with lumens 14 and 16. In some cases,distal ends 34 and 35 can have a lip adapted to use lumens 14 and 16 asinflow or outflow lumens. For example, a lip of distal ends 34 and 35can be blunt, tapered, or spoon shaped.

In some cases, a feature that permits placement of distal end 34 in thecardiovascular system can be different from a feature that permitsplacement of distal end 35 in the cardiovascular system. For example,distal end 34 can be flexible to permit manipulation within thecardiovascular tissue, whereas distal end 35 can be rigid. In somecases, distal end 35 can be adapted for use as an inflow lumen anddistal end 34 can be adapted for use as an outflow lumen. In some cases,distal end 34 can have an opening along distal region 32 transverse tolumen 14. A suitable opening can be several slits to permit blood flow,for example.

In some cases, distal end 34 can include a sensor. A suitable sensor caninclude pressure sensors or thermisters, for example. There are manydifferent cannula tip geometries that are well known to a person ofordinary skill in the art. Any one of these tip geometries may be used.Any description within this document of a particular tip geometry is notintended to be limiting, but is given merely for illustrative purposes.

In further reference to FIG. 1, regions 18, 32, and 26 of housing 10 canhave any suitable size or shape for use with a blood pump. For example,the length of any of regions 18, 32, and 26 can depend on an applicationof cannula 11. Suitable applications for cannula 11 can be providingblood flow for a blood pump recipient of any age and size, at any lengthrequired to connect a recipient's heart and a blood pump. For example,cannula 11 can be implanted in an average adult. In some cases, distalregion 32 can be between about 3 cm and about 12 cm in length for use inan average adult. For example, intermediate region 26 can be betweenabout 5 cm and about 20 cm (or even up to about 50 cm if, for example,the cannula is femorally inserted) in length when cannula 11 is usedwith a blood pump implanted in the abdominal cavity of an average adult.For example, cannula 11 can be implanted in the body of a child. In somecases, the length of proximal region 18 can be less than about 5 cm whencannula 11 is implanted in the body of a child. In some cases, thelength of intermediate region 26 can be less than about 5 cm, whencannula 11 is implanted in a child. In some cases, the length of distalregion 32 can be less than about 3 cm, if cannula 11 is implanted in achild.

Suitable applications for cannula 11 can be providing blood flow inconjunction with any blood pump. For example, cannula 11 can be usedwith an external blood pump or an implanted blood pump. In some cases,proximal region 18 can be greater than 20 cm in length when cannula 11is attached to an external blood pump. In some cases, proximal region 18can be between about 5 cm and about 20 cm in length, when cannula 11 isused with an implanted VAD. For example, intermediate region 26 can belonger than 20 cm for use with a blood pump implanted in a leg of arecipient.

Suitable applications for cannula 11 can require surgical orpercutaneous placement of cannula 11. For example, distal region 32 canbe surgically inserted through the lowest superficial part of a heart(apex) and extended across the aortic valve or placed from a peripheralartery, by crossing the aortic valve in a retrograde fashion. In somecases, the length of distal region 32 can depend on the approach thesurgeon uses to connect cannula 11 to the cardiovascular system. Forexample, for certain surgical placements, distal region 32 can begreater than 12 cm in length.

In some cases, housing 10 can have a single outer wall 28 that houseslumens 14 and 16 in the intermediate region 26 and proximal region 18and lumen 14 in distal region 32. The diameter of outer wall 28 candepend on the blood flow requirements of a particular recipient. Forexample, the diameter of outer wall 28 can be less than about 5 mm ifcannula 11 is implanted in a child. When cannula 11 is placed in anaverage adult, for example recipient, the diameter of outer wall 28 canbe between about 5 mm and about 22 mm. If cannula 11 is placed in alarge adult recipient, for example, the diameter of outer wall 28 can begreater than 22 mm.

In some cases, housing 10 can have outer walls associated with threeareas of the housing. For example, a bifurcated housing has three areas(e.g., a first branch area, a second branch area, and a third area onthe housing before the bifurcation). The first branch area has a firstouter wall 20, the second branch area has a second outer wall 22, andthe third area of the housing has an outer wall 28. Outer wall 28 canencompass intermediate region 26 and distal region 32. In some cases,first outer wall 20 can house lumen 14 in proximal region 18, and secondouter wall 22 can house lumen 16 in proximal region 18. In some cases,housing 10 can have fork 30. For example, fork 30 can provide atransition from intermediate region 26 and proximal region 18 bydividing outer wall 28 into outer walls 20 and 22.

Single outer wall 28 can have any suitable size or shape for use incannula 11. For example, outer wall 28 can be generally cylindrical(e.g., including cylinders having the cross-sectional shape of an oval,a circle, or a convex polygon, such as an octagon, a nonagon or adecagon). In some cases, the shape of outer wall 28 can contribute tothe flexibility of cannula 11. For example, outer wall 28 can be rigidor flexible. In some cases, outer wall 28 that houses proximal region 18can be adapted to engage a blood pump. For example, outer wall 28 can bebonded or compressed onto a separate rigid fitting that has a lip thatcan be held by connectors of a blood pump. In some cases, outer wall 28can be furcated to allow distal end 35 to be positioned independently oflumen 14 in intermediate region 26.

Outer walls 20 and 22 can have any appropriate size and shape. Forexample, the length of outer walls 20 and 22 can be the same as thelength of proximal region 18. If inflow and outflow ports on a bloodpump are sufficiently close, outer walls 20 and 22 can be less thanabout 5 cm in length, for example. In some cases, outer walls 20 and 22can have multiple segments, which may be areas in which proximal region18 is compressed and folded back on itself, to permit for articulationof proximal region 18. In some cases, outer wall 20 or 22 can be flaredat one end. In some cases, outer walls 20 and 22 can be adapted forengaging a blood pump as discussed for outer wall 28.

Any appropriate material for the manufacture of cannula 11 can be usedto construct outer walls 28, 20, and 22. Examples include, withoutlimitation, silicone rubbers, ethylene vinyl acetate, polyurethanes,polyether polyester copolymers, polyvinyl chloride, polyether blockamide, and polypropylene oxide. In some cases, outer walls 28, 20, and22 can be formed in whole or in part from one material or a combinationof materials. In some cases, outer walls 28, 20, and 22 can bemanufactured from different materials.

In some cases, the surfaces of housing 10 can be treated to optimizeperformance of cannula 11 in the body of a blood pump recipient. Forexample, surfaces of housing 10 can be textured or coated. In somecases, outer walls 28, 20 and 22 and the surface of the lumens 14 and 16can be treated with a layer of textured silicone. In some cases, outerwalls 28, 20, and 22 can be roughened by abrasion. In some cases, outerwalls 28, and 22 and the surface of the lumens 14 and 16 of can becoated with an antithrombotic (e.g., heparin or heparan sulfate), ananti-coagulant (e.g., bishydroxy-coumarin or warfarin) or ananti-platelet (e.g., ticlopidine or clopidogrel) agent. In some cases,the surfaces of housing 10 can have a combination of textured and coatedsurfaces. For example, housing 10 can feature textured surfaces on outerwalls 28, 20, and 22, and coated surfaces in lumens 14 and 16.

Referring to FIG. 2, a cross-section of intermediate region 26 candefine lumens 36 and 37. Lumens 36 and 37 can represent a cross-sectionof lumens 14 and 16. For example, if the cross-sectional shape of lumen14 is generally circular, then the cross-sectional shape of lumen 16 canbe generally reniform. In some cases, if the cross-sectional shape oflumen 14 has a generally reniform shape, then the cross-sectional oflumen 16 can have a generally circular shape.

Lumens 36 and 37 can have any appropriate size and shape for providing apath for blood flow. In some cases, lumen 36 can have a generallycircular cross-sectional shape (e.g., a circle, an oval or a convexpolygon, such as a decagon, a dodecagon and a tetradecagon). In somecases, lumen 37 can have a generally reniform, or kidney-like,cross-sectional shape. For example, lumen 37 can feature a notch and canbe circular or roughly circular. In some cases, the cross-sectionalshape of lumen 37 can be bilaterally symmetrical. For example, there canbe a longitudinal plane over which the reflection image of a half oflumen 37 is another half of lumen 37.

The size and shape of lumen 37 can be any size and shape that permitsblood flow. In some cases, lumen 37 can have rounded corners. Forexample, the radius of the rounded corners can be greater than about tenpercent of the radius of the largest circle that can be inscribed withinlumen 37. In some cases, the width of lumen 37 can depend on thediameter of the cross-section of intermediate region 26. For example,the diameter of the largest circle that can be inscribed in lumen 37 andthe minimum diameter of intermediate region 26 can relate to each otherby a ratio between about 0.2 and about 0.6. In another example, thediameter of the largest circle that can be inscribed in lumen 37 and theminimum inside diameter of lumen 36 can relate to each other by a ratiobetween about 0.25 and about 1.5. The geometric relationships that giverise to these dimensions are depicted in FIG. 4 and are described indetail below.

A cross-sectional view of intermediate region 26 can include septum 38located between lumens 36 and 37. In some cases, septum 38 is configuredto form convex surface 44 of lumen 37 and concave surface 48 of lumen36. In some cases, septum 38 can be configured to support the pressureproduced across septum 38 through wall tension. In some cases, septum 38can be made very thin. For example, the septum 38 can be between about0.1 mm and about 2.0 mm thick. In some cases, septum 38 can be flexible.For example, the shape of septum 38 can demonstrate directionalflexibility as compared to a flat septum. For example, septum 38 canpermit cannula 11 to bend in the plane of septum 38. In someimplementations, the septum 38 may be defined to include any materialbetween the two lumens.

Intermediate region 26 can have a malleable wire 42. In some cases,malleable wire 42 can be threaded into intermediate region 26. Forexample, malleable wire 42 can be located adjacent to lumens 14 and 16.In some cases, malleable wire 42 can extend along distal region 32 topermit a surgeon to manipulate distal end 34 through a heart and intothe vasculature of a blood pump recipient.

Any suitable material can be used to construct malleable wire 42including, medical grade stainless steel, such as an SS 303 or SS 304stainless steel, for example. Other malleable materials besidesstainless steel also may be used in the construction of the wire. Insome cases, malleable wire 42 can have a diameter that permits the wireto be easily shaped by hand into a desired configuration yet hold itsshape while housing 10 is manipulated in the recipient during placement.For example, malleable wire 42 can have stiffness about 28×10⁶ psi asdefined by ASTM D747 Standard Test Method. In some cases, theappropriate diameter of malleable wire 42 can depend on the size ofhousing 10 and wire material. In some cases, the diameter of malleablewire 42 can be from about 0.36 mm (0.014 in) to about 1.6 mm (0.063 in).In some implementations, the malleable wire 42 need not have anyparticular cross-section. For example, the wire 42 may be formed as acoil.

In further reference to FIG. 2, intermediate region 26 can define athird lumen 40 and a fourth lumen 43. In some cases, lumens 40 and 43can be located between lumen 36 and the outer wall 28, below the areadefined by the rounded corners of lumen 37. For example, lumen 40 can belocated on one side of lumen 36, and lumen 43 can be located on theopposite side of lumen 36. In some cases, lumens 40 and 43 can havegenerally circular cross-sectional shapes, including ovals, circles, andpolygons. In some cases, the cross-sectional shape of lumen 40 candiffer from the cross-sectional shape of lumen 43. In some cases, thecenters of lumens 40 and 43 can lie on the axis that transects lumen 36along its diameter and is perpendicular to a bilateral axis of the lumen37.

Lumens 40 and 43 can perform any appropriate function in cannula 11. Insome cases, lumens 40 and 43 can be used to minimize structuralrigidity. For example, lumens 40 and 43 can be hollow to reduce therigidity of intermediate region 26. In some cases, lumen 40 can be usedto provide local access to the circulatory system. For example, lumen 40can include a fluid port. A fluid port placed in intermediate region 26can provide local access to the circulatory system for purposes ofpressure measurement, blood sampling, or fluid administration. In somecases, the intermediate region or lumen 40 can have a fluid port that isadapted to deliver fluid into the circulatory system or enable a user towithdraw blood from a blood pump recipient.

In certain cases, the intermediate region and/or lumen 40 can beutilized to introduce a sensor into the circulatory system. For example,lumen 40 can be located in intermediate region 26 specifically to housea probe or sensor. An appropriate location for a sensor can bedetermined by probe or sensor type. For example, suitable probes orsensors can be mechanical, piezoelectric, fiber optic, ultrasonic, ormicroelectro-mechanical probes or sensors. In some cases, a probe orsensor can be a single probe or sensor or a combination of probes orsensors. In some cases, a probe or sensor can be used to providereal-time information about blood flow or temperature. For example, aprobe or sensor can be a pressure transducer, a flow sensor, or athermister. In certain cases, the intermediate region can be adapted forrouting a sensor or sensor lead, sensor wire, or an electrode attachedto a sensor.

In some implementations, the sensor is not in the fluid port as directblood contact is not necessary for certain sensors. In such cases, thelumen 40 may house the sensor and wires. Referring to FIG. 3,subsequently described, the location of a sensor could be anywhere alongcannula 11. For example, a sensor could be placed at a positionassociated with distal end 35. The position of the sensor may depend onthe type of sensor and what is being measured.

FIG. 3 depicts an exemplary embodiment of a cannula 11 for use with ablood pump 12. Blood pump 12 can be one of any number of blood pumpsavailable for providing mechanical circulatory support. For example, ablood pump can be implanted or implantable in a blood pump recipient. Insome cases, a blood pump can be an external pump, such as acardiopulmonary bypass pump. A blood pump can be a ventricular assistdevice (VAD). For example, depending on the placement of the cannula, ablood pump can be a left ventricular assist device (LVAD) or a rightventricular assist device (RVAD). Additionally, the blood pump can serveas a bilateral ventricular assist device (BiVAD). In some cases, a bloodpump can be a continuous-flow blood pump or a pulsatile-flow blood pump.

Referring to FIG. 4, the size of lumen 36, and the shape and size oflumen 37 can determine the flow characteristics associated with cannula11. In some cases, the shape of lumen 37 can be limited by the size ofintermediate region 26 and lumen 36.

As depicted in FIG. 4, lumen 36 has radius (R₁) 52, a cross-section ofintermediate region 26 has radius (R₂) 54, and circle 50 inscribed in arounded corner of lumen 37 has radius (R₃) 56. Triangle 58 (t), withvertices at the centers of lumen 36, intermediate region 26 and circle50, has angles α 60, β, and γ, and sides a, b, and c. The area (A_(t))of triangle 58 can be described using basic trigonometry relations:

(Law of Cosines)c ² =a ² +b ²−2·a·b·cos(γ)

(Law of Sines)

$\frac{a}{\sin(\alpha)} = {\frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}}$So:$\gamma = {{acos}\left( \frac{a^{2} + b^{2} - c^{2}}{2{a \cdot b}} \right)}$$\beta = {{asin}\left( {\frac{b}{c} \cdot {\sin(\gamma)}} \right)}$$\alpha = {{\sin\left( {\frac{a}{c} \cdot {\sin(\gamma)}} \right)} = {\pi - \beta - \gamma}}$$A_{t} = {{\frac{1}{2} \cdot a \cdot c \cdot {\sin(\beta)}} = {{\frac{1}{2} \cdot a \cdot b \cdot {\sin(\gamma)}} = {\frac{1}{2} \cdot b \cdot c \cdot {\sin(\alpha)}}}}$Relating the triangle sides to the cannula:a=R ₂ −R ₃b=R ₂ −R ₁c=R ₁ +R ₃

$\gamma = {{acos}\left\lbrack \frac{\left( {R_{2} - R_{3}} \right)^{2} + \left( {R_{2} - R_{1}} \right)^{2} - \left( {R_{1} + R_{3}} \right)^{2}}{2{\left( {R_{2} + R_{3}} \right) \cdot \left( {R_{2} - R_{1}} \right)}} \right\rbrack}$$\beta = {{asin}\left( {\frac{R_{2} - R_{1}}{R_{1} + R_{3}} \cdot {\sin(\gamma)}} \right)}$α = 180 − γ − βThe area defined by the equation:

$A_{t} = {\frac{1}{2}\left( {R_{2} - R_{3}} \right)\left( {R_{1} + R_{3}} \right){\sin(\beta)}}$

Further simplifying (not carrying the extra angle):

$\beta = {{asin}\left\lbrack {\frac{R_{2} - R_{1}}{R_{1} + R_{3}} \cdot {\sin\left\lbrack {{acos}\left\lbrack \frac{\left( {R_{2} - R_{3}} \right)^{2} + \left( {R_{2} - R_{1}} \right)^{2} - \left( {R_{1} + R_{3}} \right)^{2}}{2{\left( {R_{2} - R_{3}} \right) \cdot \left( {R_{2} - R_{1}} \right)}} \right\rbrack} \right\rbrack}} \right\rbrack}$The perimeter (S₀) of lumen 37 can be determined by:S _(O)=2·[(π−γ)·R ₂+(π−β)·R ₃ +α·R ₁]The area of lumen 37 (A_(o)) is given by:

$\begin{matrix}{A_{O} = {{\left( {{\pi \cdot R_{2}^{2}} - {\pi \cdot R_{1}^{2}}} \right)\ldots}\mspace{11mu} + {{- 2} \cdot}}} \\{\left( {{\frac{\gamma}{2 \cdot \pi} \cdot \pi \cdot R_{2}^{2}} - {\frac{\pi - \alpha}{2 \cdot \pi} \cdot \pi \cdot R_{1}^{2}} - A_{t} - {\frac{\pi - \beta}{2 \cdot \pi} \cdot \pi \cdot R_{3}^{2}}} \right)} \\{= {{\left( {\pi - \gamma} \right)R_{2}^{2}} - {R_{1}^{2}\alpha} + {\left( {\pi - \beta} \right)R_{3}^{2}} + {2A_{t}}}}\end{matrix}$

The relation between radius 56 (R₃) and angle α 60 can be found from thecosine law:

${\cos\;\alpha} = \frac{\left( {R_{1} + R_{3}} \right)^{2} + \left( {R_{2} - R_{1}} \right)^{2} - \left( {R_{2} - R_{3}} \right)^{2}}{2\left( {R_{1} + R_{3}} \right)\left( {R_{2} - R_{1}} \right)}$Expanding gives:

${2\;\cos\;\alpha} = \frac{\begin{matrix}{R_{1}^{2} + R_{3}^{2} + {2R_{1}R_{3}} + R_{2}^{2} + R_{1}^{2} -} \\{{2R_{1}R_{2}} - R_{2}^{2} - R_{3}^{2} + {2R_{2}R_{3}}}\end{matrix}}{{R_{1}R_{2}} - R_{1}^{2} + {R_{3}\left( {R_{2} - R_{1}} \right)}}$Simplifying and rearranging yields:2(R ₁ R ₂ −R ₁ ²)cos α+2R ₃(R ₂ −R ₁)cos α=2R ₁ ²−2R ₁ R ₂+2R ₃(R ₁ +R₂)2(R ₁ R ₂ −R ₂)cos α−2R ₁ ²+2R ₁ R ₂=2R ₃(R ₁ +R ₂−(R ₂ −R ₀)cos α)

$R_{3} = \frac{{{R_{1}\left( {R_{2} - R_{1}} \right)}\cos\;\alpha} + {R_{1}\left( {R_{2} - R_{1}} \right)}}{R_{1} + R_{2} - {\left( {R_{2} - R_{1}} \right)\cos\;\alpha}}$$R_{3} = \frac{{R_{1}\left( {R_{2} - R_{1}} \right)}\left( {1 + {\cos\;\alpha}} \right)}{\left( {R_{1} + R_{2}} \right) - {\left( {R_{2} - R_{1}} \right)\cos\;\alpha}}$With R₂=1, values for radius 56 at angle α 60 from 40° to 140° andradius R₁ are listed in table 1 below.

TABLE 1 Angle α (degree) R₁ 40 50 60 70 80 90 100 110 120 130 140(degree/radian) 0.6981 0.8727 1.0472 1.2217 1.3963 1.5708 1.7453 1.91992.0944 2.2689 2.4435 0.250 0.4902 0.4011 0.3214 0.2533 0.1965 0.15000.1123 0.0819 0.0577 0.0387 0.0240 0.275 0.4893 0.4049 0.3277 0.26050.2036 0.1564 0.1176 0.0861 0.0609 0.0409 0.0255 0.300 0.4856 0.40580.3316 0.2657 0.2091 0.1615 0.1221 0.0898 0.0636 0.0429 0.0268 0.3250.4795 0.4044 0.3332 0.2691 0.2132 0.1656 0.1257 0.0928 0.0660 0.04460.0279 0.350 0.4715 0.4009 0.3329 0.2707 0.2158 0.1685 0.1285 0.09520.0679 0.0460 0.0288 0.375 0.4619 0.3956 0.3309 0.2709 0.2172 0.17050.1306 0.0971 0.0694 0.0471 0.0296 0.400 0.4507 0.3887 0.3273 0.26960.2174 0.1714 0.1319 0.0984 0.0706 0.0480 0.0302 0.425 0.4384 0.38040.3223 0.2670 0.2164 0.1715 0.1324 0.0991 0.0713 0.0486 0.0306 0.4500.4249 0.3708 0.3160 0.2632 0.2144 0.1707 0.1323 0.0994 0.0717 0.04900.0309 0.475 0.4105 0.3601 0.3085 0.2584 0.2115 0.1691 0.1316 0.09920.0718 0.0492 0.0311 0.500 0.3953 0.3485 0.3000 0.2525 0.2076 0.16670.1302 0.0984 0.0714 0.0490 0.0311 0.525 0.3793 0.3359 0.2905 0.24560.2029 0.1635 0.1282 0.0972 0.0707 0.0487 0.0309 0.550 0.3627 0.32250.2802 0.2379 0.1973 0.1597 0.1256 0.0956 0.0697 0.0481 0.0306 0.5750.3454 0.3084 0.2690 0.2294 0.1910 0.1552 0.1225 0.0935 0.0684 0.04720.0301 0.600 0.3277 0.2936 0.2571 0.2201 0.1840 0.1500 0.1188 0.09090.0667 0.0462 0.0294 0.625 0.3094 0.2782 0.2446 0.2102 0.1763 0.14420.1146 0.0880 0.0647 0.0449 0.0287 0.650 0.2907 0.2623 0.2314 0.19950.1680 0.1379 0.1099 0.0846 0.0623 0.0433 0.0277

Assuming incompressible and fully developed steady flow in a straighttube, the Navier Stoke's equation can be simplified to Poisson'sequation.

$0 = {{- \frac{\partial P}{\partial z}} + {\mu\left( {\frac{\partial^{2}w}{\partial x^{2}} + \frac{\partial^{2}w}{\partial y^{2}}} \right)}}$

LetX=x/R ₂Y=y/R ₂W=w/V _(ref)

Where R₂ 54 is the radius of intermediate region 26 and V_(ref) is themean flow velocity in circular pipe of radius R₂.

Use Poiseulle flow as reference, we have:

$Q_{ref} = {\frac{\pi\; R_{2}^{4}}{8\mu}\left( {- \frac{\partial P}{\partial z}} \right)}$$V_{ref} = {\frac{Q_{ref}}{\pi\; R_{2}^{2}} = {\frac{R_{2}^{2}}{8\;\mu}\left( {- \frac{\partial P}{\partial z}} \right)}}$

Substitute into the Poisson equation gives the non-dimensional form,

$\begin{matrix}{{- 8} = \left( {\frac{\partial^{2}W}{\partial X^{2}} + \frac{\partial^{2}W}{\partial Y^{2}}} \right)} & (1)\end{matrix}$

For wall shear rate, we have:

$\tau = {\mu\frac{\mathbb{d}w}{\mathbb{d}n}}$

Where n is the direction normal to the wall. In non-dimensional form, itis:

$\frac{\mathbb{d}w}{\mathbb{d}n} = {\frac{V_{ref}}{R_{2}}\left( \frac{\mathbb{d}W}{\mathbb{d}N} \right)}$

This simplifies to:

$\begin{matrix}{\frac{\mathbb{d}w}{\mathbb{d}n} = {\frac{R_{2}}{8\mu}\left( {- \frac{\partial P}{\partial z}} \right)\left( \frac{\mathbb{d}W}{\mathbb{d}N} \right)}} & (2)\end{matrix}$

Solving (1) numerically with appropriate boundary equations we obtainthe non-dimensionalized values of velocity W from which we can calculatethe non-dimensional flow rate Q*, wall shear rate dW/dn, area A*. Thedimensional values can be evaluated by the following conversionrelations:

$\begin{matrix}{\frac{\mathbb{d}w}{\mathbb{d}n} = {\frac{\mathbb{d}W}{\mathbb{d}n}\frac{R_{2}}{8\mu}\left( {- \frac{\partial P}{\partial z}} \right)}} & (3) \\{Q = {{Q^{*}R_{2}^{2}V_{ref}} = {Q^{*}\frac{R_{2}^{4}}{8\mu}\left( {- \frac{\partial P}{\partial z}} \right)}}} & (4) \\{A = {A^{*}R_{2}^{2}}} & (5) \\{w_{\max} = {{W_{\max}V_{ref}} = {W_{\max}\frac{R_{2}^{2}}{8\mu}\left( {- \frac{\partial P}{\partial z}} \right)}}} & (6)\end{matrix}$

The solution for the fluid dynamics of a conduit with cross-sectionsdefined by lumens 36 and 37 can be converted to dimensional units asfollows:

-   -   1 Poise=1 gm/cm-sec    -   1 cP=0.01 Poise=0.01 gm/cm-sec    -   1 Pa=1 N/m²=1 kg-m/sec²-m²=10 gm/cm-sec²    -   1 mmHg=0.1333223684211 kPa=1333.223684211 gm/cm-sec²    -   1 litre/min=1000 cm³/min=16.67 cm³/sec

Therefore, if the pressure gradient (∂P/∂z) is expressed in mmHg/cm, theshear rate (γ) in sec⁻¹, the viscosity (μ) in cP, the velocity (w) incm/sec, the flow rate (Q) in 1/min, the dimensions x and y in cm, thediameter (D), the radius (r) in cm, and the area (A) in cm², Poisson'sequation, can be expressed in dimensional form, as follows:

${{\left( \frac{\partial P}{\partial z} \right) \times 1333.2236843\mspace{14mu}{gm}\text{/}{cm}^{2}} - \sec^{2}} = {{\mu \times 0.01\mspace{14mu}{gm}\text{/}{cm}} - {\sec \times \left( {\frac{\partial^{2}w}{\partial x^{2}} + \frac{\partial^{2}w}{\partial y^{2}}} \right)\mspace{11mu}{cm}\text{/}\sec} - {cm}^{2}}$Giving

${\frac{1}{\mu}\left( \frac{\partial P}{\partial z} \right) \times 133322.36842} = \left( {\frac{\partial^{2}w}{\partial x^{2}} + \frac{\partial^{2}w}{\partial y^{2}}} \right)$

For conversion from non-dimensional values to actual values:

Wall shear rate calculation, the factor

$\frac{R_{2}}{8\mu}\left( {- \frac{\partial P}{\partial z}} \right)$will be,

${\frac{R_{2}}{8\mu}\frac{cm}{{0.01\mspace{11mu}{gm}\text{/}{cm}} - \sec}\left( {- \frac{\partial P}{\partial z}} \right) \times 1333.22368\frac{gm}{{cm} - \sec^{2}}\frac{1}{cm}} = {\frac{1333.22368}{8 \times 0.01}\frac{R_{2}}{\mu}\left( {- \frac{\partial P}{\partial z}} \right)}$OR$\frac{\mathbb{d}w}{\mathbb{d}h} = {\frac{\mathbb{d}W}{\mathbb{d}n} \times 16665.29605\frac{R_{2}}{\mu}\left( {- \frac{\partial P}{\partial z}} \right)\sec^{- 1}}$Flow rate calculation will be,

$\frac{1}{16.67}\frac{1}{{cm}^{3}/\sec}\frac{R_{2}^{4}}{8\mu}\frac{{cm}^{4}}{{0.01\mspace{11mu}{{gm}/{cm}}} - \sec}\left( {- \frac{\partial P}{\partial z}} \right) \times 1333.22368\frac{gm}{{cm} - \sec^{2}}\frac{1}{cm}$OR$Q = {Q^{*} \times 999.7178 \times \frac{R_{2}^{4}}{\mu}\left( {- \frac{\partial P}{\partial z}} \right){1/\min}}$Area calculation is dependent on actual value of radius (R₂) 54 ofintermediate region 26, such that,A=A*R ₂ ² cm²Velocity calculation will have the same factor as shear rate.

$w_{\max} = {W_{\max} \times 16665.29605\frac{R_{2}^{2}}{\mu}\left( {- \frac{\partial P}{\partial z}} \right)\mspace{11mu}{cm}\text{/}\sec}$

For Poiseulle flow,

$Q = {\frac{\pi\; D^{4}}{128\;\mu}\left( {- \frac{\partial P}{\partial z}} \right)}$Average flow velocity V_(ave) is

$V_{ave} = \frac{Q}{A}$Maximum velocity V_(max) is V_(max)=2V_(ave)Shear rate γ at radius r is

$\gamma = {\frac{r}{2\;\mu}\left( {- \frac{\partial P}{\partial z}} \right)}$

Therefore,

$\frac{Q \times 16.67^{c\; m^{3}}}{\sec} = {\frac{\pi \times D^{4}{cm}^{4}}{\frac{128 \times \mu \times 0.01^{gm}}{{cm} - \sec}} \times \frac{1333.22368^{gm}}{{cm} - {\sec^{2}\frac{1}{cm}\left( {- \frac{\partial P}{\partial z}} \right)}}}$Giving

$\begin{matrix}{{\frac{Q \times \mu}{196.3333937 \times D^{4}} = {\left( {- \frac{\partial P}{\partial z}} \right)\mspace{14mu}{Or}}}{\frac{Q \times \mu}{3141.3343 \times r^{4}} = {\left( {- \frac{\partial P}{\partial z}} \right)\mspace{14mu}{mmHg}\text{/}{cm}}}} & (12)\end{matrix}$And

$\gamma = {\frac{r}{2 \times \mu \times 0.01}\frac{cm}{\frac{gm}{{cm} - \sec}}\left( {- \frac{\partial P}{\partial z}} \right) \times 1333.22368\frac{gm}{{cm} - \sec^{2}}\frac{1}{cm}}$Giving

$\begin{matrix}{\gamma = {66661.1842\frac{r}{\mu}\left( {- \frac{\partial P}{\partial z}} \right)\mspace{14mu}\sec^{- 1}}} & (13)\end{matrix}$

The maximum velocity is, using equation (12),

$\begin{matrix}{V_{\max} = {\frac{2 \times 1000}{60\;\pi\; r^{2}} \times \frac{3141.3343\; r^{4}}{\mu}\left( {- \frac{\partial P}{\partial z}} \right)}} & \; \\{V_{\max} = {33330.6\frac{r^{2}}{\mu}\left( {- \frac{\partial P}{\partial z}} \right){cm}\text{/}\sec}} & (14)\end{matrix}$

For eccentric annulus flow, the theoretical formula for eccentricannulus flow is given by White (Viscous Fluid Flow, 1974, McGraw-Hill,ISBN 0-07-069710, equation 3.50) and is reproduced here. The flow rate Qis:

$Q = {\frac{\pi}{8\;\mu}{\left( {- \frac{\partial P}{\partial z}} \right)\left\lbrack {a^{4} - b^{4} - \frac{4\; c^{2}M^{2}}{\beta - \alpha} - {8\; c^{2}M^{2}{\sum\limits_{n = 1}^{\infty}\;\frac{n\;{\mathbb{e}}^{- {n{({\beta + \alpha})}}}}{\sinh\left( {{n\;\beta} - {n\;\alpha}} \right)}}}} \right\rbrack}}$where M = (F² − a²)^(1/2) $F = \frac{a^{2} - b^{2} + c^{2}}{2\; c}$$\alpha = {\frac{1}{2}\ln\frac{F + M}{F - M}}$$\beta = {\frac{1}{2}\ln\frac{F - c + M}{F - c - M}}$and a is equal to R₂ 54, b is equal to R₁ 52 in the other equations.

Note that in this equation, when b=0 then c=a, giving M=0. This isreduced to the Poiseulle equation. The resultant constant in theequation is same as in (12) above, so the flow rate is:

$\begin{matrix}{Q = {\frac{3141.3343}{\mu}{\left( {- \frac{\partial P}{\partial z}} \right)\left\lbrack {a^{4} - b^{4} - \frac{4\; c^{2}M^{2}}{\beta - \alpha} - {8\; c^{2}M^{2}{\sum\limits_{n = 1}^{\infty}\;\frac{n\;{\mathbb{e}}^{- {n{({\beta + \alpha})}}}}{\sinh\left( {{n\;\beta} - {n\;\alpha}} \right)}}}} \right\rbrack}\mspace{14mu} l\text{/}\min}} & (15)\end{matrix}$

Q can be expressed in non-dimensional form as well if divide Q bycorresponding Poiseulle flow Q_(ref) and let a=a′R₂, b=b′R₂, c=c′R₂,M=M′R₂ ². This gives:

$Q_{nd} = {{\frac{\frac{\pi}{8\;\mu}\left( {- \frac{\partial P}{\partial z}} \right)}{\frac{\pi\; R_{2}^{4}}{8\;\mu}\left( {- \frac{\partial P}{\partial z}} \right)}\left\lbrack {{a^{\prime 4}R_{2}^{4}} - {b^{\prime 4}R_{2}^{4}} - \frac{4\; c^{\prime 2}R_{2}^{2}M^{\prime 2}R_{2}^{2}}{\beta - \alpha} - {8\; c^{\prime 2}R_{2}^{2}M^{\prime 2}R_{2}^{2}{\sum\limits_{n = 1}^{\infty}\;\frac{n\;{\mathbb{e}}^{- {n{({\beta + \alpha})}}}}{\sinh\left( {{n\;\beta} - {n\;\alpha}} \right)}}}} \right\rbrack}\mspace{14mu}{or}}$$Q_{nd} = \left\lbrack {a^{\prime 4} - b^{\prime 4} - \frac{4\; c^{\prime 2}M^{\prime 2}}{\beta - \alpha} - {8\; c^{\prime 2}M^{\prime 2}{\sum\limits_{n = 1}^{\infty}\;\frac{n\;{\mathbb{e}}^{- {n{({\beta + \alpha})}}}}{\sinh\left( {{n\;\beta} - {n\;\alpha}} \right)}}}} \right\rbrack$Since Q_(nd) is obtained by normalizing to Q_(ref), the relation to Q*is Q*=πQ_(nd) as the limiting value of Q_(nd) is 1.

This document also provides methods for implanting a multi-lumen cannulainto the heart of a mammal. In some cases, a distal end of the outflowlumen can be positioned a blood vessel, such as the pulmonary artery orthe aorta. In some cases, distal end of the inflow lumen can bepositioned in a chamber of the heart, such as the left ventricle or theright atrium.

For example, when used with LVADs, the distal region of a cannula can beinserted through a single puncture site in the lowest superficial partof the heart (apex) and extended across the aortic valve. Puncturing theapex of heart can be accomplished by any appropriate method (e.g.,cannulation, incision, or excision of the myocardium). In some cases, acannula can be positioned such that the distal end of the second lumenis in the left ventricle and the distal end of the first lumen is in theaorta. A cannula provided herein can be connected to a heart byanastomosis. For, example, once the cannula is in the desired positionin the heart, the cannula can be secured to the myocardium with sutures.A similar arrangement can be used for right heart support.

Referring to an example implementation shown in FIG. 5, cannula 11 canbe inserted into the cardiovascular system, for providing right-sidesupport to a recipient's heart 62 in conjunction with pump 12. Forexample, cannula 11 can provide an inflow path for blood from arecipient's right ventricle 64 and an outflow path for blood exitingpump 12 to the pulmonary artery 68. In some cases, pump 12 can be placedin the abdominal cavity of a recipient, and be connected to an externalpower supply by drive-line 76. Of course, although not shown in FIG. 5,the cannula 11 also may be inserted to provide left-side support to theheart 62 (e.g., so that blood from the left ventricle is directed to theblood pump, and pumped into the aorta).

Other configurations for surgical placement of a multi-lumen cannula canbe utilized. For example, a multi-lumen cannula can be placed from aperipheral artery by crossing the aortic valve in a retrograde fashion.In some cases, right ventricle support can be similarly achieved bypassing a cannula across both right-sided valves in an antigradefashion. In some cases, a transceptal approach can be utilized bypositioning the distal end of the second lumen in the left atrium orleft ventricle and the distal end of the first lumen in the aorta.

The invention will be further described in the following examples, whichdo not limit the scope of the invention described in the claims.

EXAMPLES Example 1 Flow characteristics

The flow characteristics for the geometry described in FIG. 4 have beencomputed in a non-dimensionalized form and converted to dimensionalvalues in general range of those expected for pumping blood. Using theconfiguration presented in FIG. 4, with radius 52, R₁=0.3125 cm, andangle 60, α=80°, the pressure gradient was expressed in mmHg/cm,viscosity in cP, flow rate in l/min, velocity in cm/sec and alldimensional measures in cm, cm². The pressure gradient per unit lengthmay be selected arbitrarily and the results can be linearly scaled tomatch the capacity of the blood pump and physiologic configurationappropriately.

Where

-   -   viscosity μ=4 cP    -   Pressure gradient ∂P/∂z=0.01 mmHg/cm    -   R₂ 54=0.5 cm

The non-dimensional values were calculated using numerical methods suchas finite element method, and the actual values for lumen 37 weredetermined to be:

${{Flow}\mspace{14mu}{rate}} = {{0.42383 \times 999.7178 \times \frac{0.5^{4}}{4} \times 0.01} = {0.0662\mspace{14mu} l\text{/}\min}}$$\begin{matrix}{{{Maximum}\mspace{14mu}{wall}\mspace{14mu}{shear}\mspace{14mu}{rate}} = {3.24401 \times 16665.296 \times \frac{0.5}{4} \times 0.01}} \\{= {67.6\mspace{14mu}\sec^{- 1}}}\end{matrix}$ $\begin{matrix}{{{Minimum}\mspace{14mu}{wall}\mspace{14mu}{shear}\mspace{14mu}{rate}} = {0.85740 \times 16665.296 \times \frac{0.5}{4} \times 0.01}} \\{= {17.9\mspace{14mu}\sec^{- 1}}}\end{matrix}$ $\begin{matrix}{{{Maximum}\mspace{14mu}{flow}\mspace{14mu}{velocity}} = {0.52475 \times 16665.296 \times \frac{0.5^{2}}{4} \times 0.01}} \\{= {5.5\mspace{14mu}{cm}\text{/}\sec}}\end{matrix}$Cross sectional area=1.69483×0.5²=0.424 cm²

The corresponding lumen 36 data were, using equations 12 to 14:

${{Flow}\mspace{14mu}{rate}} = {{\frac{3141.334 \times 0.3125^{4}}{4} \times 0.01} = {0.075{\mspace{11mu}\;}l\text{/}\min}}$${{Wall}\mspace{14mu}{shear}\mspace{14mu}{rate}} = {{66661.1842 \times \frac{0.3125}{4} \times 0.01} = {52\mspace{14mu}\sec^{- 1}}}$${{Maximum}\mspace{14mu}{flow}\mspace{14mu}{velocity}} = {{33330.6 \times \frac{0.3125^{2}}{4} \times 0.01} = {8.14\mspace{14mu}{cm}\text{/}\sec}}$Cross sectional area=π×0.3125²=0.307 cm²

The actual values were calculated for different configurations of lumens36 and 37 to determine the relationship of changes in lumen geometry toflow rate, wall shear rate, velocity and total pressure gradient.

The data for flow rate are tabulated in table 2, with R₂=1. These datashow that flow rate increased asymptotically, and that the increase wasinsignificant when a 60 is greater than 90°. This was determined to be aproduct of the small increase in cross sectional area and low flowvelocity in that area of the cannula. The flow rate decreasedasymptotically towards zero as R₁ 52 increased.

TABLE 2 Angle α (degree) R₁ 40 50 60 70 80 90 100 110 120 130 140 0.2502.13310 2.19580 2.23110 2.25050 2.26030 2.26490 2.26720 2.26790 2.268202.26830 2.26830 0.275 1.98030 2.04840 2.08730 2.10870 2.11960 2.125002.12760 2.12840 2.12880 2.12860 2.12880 0.300 1.82710 1.90010 1.941601.96470 1.97690 1.98280 1.98560 1.98670 1.98680 1.98690 1.98700 0.3251.67608 1.75191 1.79585 1.82047 1.83340 1.83985 1.84283 1.84388 1.844221.84432 1.84455 0.350 1.52820 1.60560 1.65117 1.67662 1.69031 1.697401.70045 1.70169 1.70214 1.70226 1.70222 0.375 1.38461 1.46237 1.508581.53505 1.54928 1.55653 1.55989 1.56121 1.56169 1.56189 1.56184 0.4001.24653 1.32326 1.36963 1.39656 1.41113 1.41856 1.42203 1.42346 1.423941.42411 1.42418 0.425 1.11469 1.18947 1.23534 1.26207 1.27681 1.284301.28793 1.28947 1.29000 1.29014 1.29010 0.450 0.98981 1.06162 1.106191.13260 1.14725 1.15478 1.15842 1.16003 1.16057 1.16076 1.16072 0.4750.87240 0.94019 0.98308 1.00882 1.02320 1.03072 1.03427 1.03586 1.036511.03664 1.03668 0.500 0.76281 0.82615 0.86677 0.89143 0.90530 0.912660.91615 0.91772 0.91835 0.91856 0.91864 0.525 0.66126 0.71970 0.757610.78086 0.79416 0.80109 0.80456 0.80624 0.80684 0.80705 0.80704 0.5500.56785 0.62099 0.65595 0.67753 0.69013 0.69672 0.70005 0.70171 0.702280.70240 0.70245 0.575 0.48273 0.53045 0.56209 0.58194 0.59350 0.599740.60295 0.60443 0.60502 0.60521 0.60524 0.600 0.40575 0.44790 0.476170.49413 0.50475 0.51056 0.51349 0.51481 0.51543 0.51557 0.51567 0.6250.33682 0.37338 0.39836 0.41431 0.42383 0.42914 0.43181 0.43303 0.433650.43377 0.43388 0.650 0.27565 0.30695 0.32851 0.34246 0.35080 0.355570.35797 0.35911 0.35964 0.35982 0.35981 0.675 0.22198 0.24826 0.266540.27841 0.28579 0.28992 0.29207 0.29306 0.29351 0.29372 0.29369 0.7000.17551 0.19708 0.21227 0.22225 0.22845 0.23208 0.23390 0.23480 0.235150.23533 0.23537 0.725 0.13577 0.15311 0.16543 0.17362 0.17876 0.181790.18333 0.18409 0.18435 0.18455 0.18457 0.750 0.10239 0.11594 0.125630.13214 0.13631 0.13874 0.14000 0.14064 0.14092 0.14104 0.14107 0.7750.07488 0.08511 0.09247 0.09751 0.10078 0.10263 0.10367 0.10416 0.104420.10450 0.10455 0.800 0.05273 0.06014 0.06553 0.06928 0.07167 0.073100.07393 0.07432 0.07447 0.07452 0.07457

The maximum shear rate (Table 3) decreased linearly as R₁ 52 increased.The minimum wall shear rate (Table 4) decreased asymptotically towardszero as α 60 increased. When R₃ 56 decreased towards zero (as α 60increased towards 180) the flow velocity in the region bound by the arcof R₃ 56 also decreases to zero. Therefore, the wall shear ratedecreased to zero.

TABLE 3 Angle α (degree) R₁ 40 50 60 70 80 90 100 110 120 130 140 0.2506.57130 6.68910 6.94640 6.93310 6.83720 6.92180 6.96530 6.92200 6.971606.95410 6.93730 0.275 6.33170 6.52820 6.59760 6.55530 6.61060 6.776206.70590 6.70390 6.66730 6.77580 6.67920 0.300 6.04000 6.27790 6.394806.40620 6.37380 6.52200 6.46450 6.45490 6.44030 6.53450 6.41100 0.3255.84551 6.04709 6.14347 6.10662 6.20206 6.18873 6.21725 6.26717 6.182306.14669 6.21901 0.350 5.66895 5.83101 5.89415 5.89114 5.96762 5.891505.92645 5.87783 5.89087 5.93102 5.96582 0.375 5.45116 5.57666 5.613815.74144 5.64999 5.67678 5.70371 5.66028 5.65532 5.67529 5.66801 0.4005.17482 5.32309 5.39263 5.48418 5.42821 5.41749 5.45975 5.40620 5.389125.46658 5.43339 0.425 4.96221 5.11576 5.20807 5.23654 5.15413 5.206545.16295 5.16728 5.16251 5.16992 5.15455 0.450 4.74298 4.86049 4.955814.91659 4.95511 4.94617 4.91734 4.93898 4.91290 4.93083 4.90958 0.4754.53841 4.66292 4.66683 4.69966 4.70253 4.69435 4.69059 4.67150 4.685704.68978 4.66700 0.500 4.32062 4.42209 4.43088 4.47055 4.44652 4.463824.43656 4.45218 4.45053 4.45225 4.46200 0.525 4.10839 4.18325 4.185374.20610 4.22287 4.20109 4.20740 4.21911 4.21652 4.22435 4.20678 0.5503.86883 3.91672 3.93953 3.97136 3.97282 3.97263 3.96551 3.99208 3.975613.94352 3.98792 0.575 3.64810 3.70817 3.71502 3.72443 3.72993 3.723583.72921 3.71925 3.70600 3.73639 3.72447 0.600 3.41940 3.49302 3.423643.47994 3.47462 3.49419 3.48305 3.50749 3.50765 3.48362 3.47092 0.6253.21533 3.23356 3.21538 3.20698 3.24401 3.24110 3.19194 3.20103 3.260883.23870 3.24760 0.650 2.97994 2.95760 2.95990 2.98417 2.96070 2.969502.97194 3.00872 2.97904 2.97086 2.98150 0.675 2.72063 2.74334 2.783582.75232 2.74335 2.74414 2.75337 2.77350 2.73652 2.74651 2.76404 0.7002.53872 2.52008 2.51784 2.54512 2.53893 2.53395 2.54593 2.54784 2.534852.52988 2.51605 0.725 2.29594 2.30174 2.30170 2.29923 2.31077 2.314192.30718 2.31854 2.30629 2.30954 2.31481 0.750 2.07188 2.08666 2.087762.08728 2.08671 2.07885 2.09260 2.09292 2.09083 2.08274 2.09073 0.7751.87483 1.86542 1.86361 1.84379 1.85891 1.85818 1.88397 1.85859 1.869141.85212 1.86362 0.800 1.64411 1.64434 1.61839 1.62330 1.59794 1.626351.63324 1.63680 1.65552 1.58783 1.61462

TABLE 4 Angle α (degree) R₁ 40 50 60 70 80 90 100 110 120 130 140 0.2502.67750 2.42100 2.12910 1.81230 1.49010 1.16530 0.86129 0.59640 0.383100.22764 0.12672 0.275 2.61570 2.36500 2.08320 1.77550 1.45680 1.143700.85226 0.59367 0.38720 0.23372 0.13137 0.300 2.54320 2.29400 2.032501.72670 1.42330 1.12280 0.83804 0.59022 0.38824 0.23868 0.13680 0.3252.47399 2.23412 1.97826 1.68007 1.38501 1.10102 0.82374 0.58309 0.389680.24304 0.13981 0.350 2.39937 2.17399 1.91828 1.63875 1.35237 1.074700.80903 0.57624 0.38914 0.24545 0.14311 0.375 2.32136 2.10653 1.857891.58804 1.31309 1.04207 0.79123 0.57072 0.38701 0.24663 0.14531 0.4002.24459 2.02869 1.79561 1.53655 1.27326 1.01105 0.77581 0.56105 0.385410.24746 0.14785 0.425 2.16124 1.96242 1.73468 1.48764 1.23191 0.985980.75688 0.55261 0.38026 0.24609 0.14786 0.450 2.08042 1.88661 1.664961.43553 1.18850 0.95490 0.73694 0.53921 0.37668 0.24562 0.14819 0.4751.97856 1.81131 1.60025 1.37215 1.14431 0.91724 0.71551 0.52683 0.370370.24372 0.14788 0.500 1.90034 1.72797 1.53543 1.32912 1.10169 0.881450.69643 0.51347 0.36029 0.24076 0.14775 0.525 1.81358 1.65407 1.463961.25620 1.06201 0.85027 0.66734 0.49688 0.35170 0.23645 0.14558 0.5501.71489 1.56912 1.39270 1.19935 1.00575 0.81053 0.64458 0.48123 0.343800.23132 0.14350 0.575 1.63077 1.48768 1.31922 1.14336 0.96022 0.771330.61670 0.46341 0.33313 0.22452 0.14043 0.600 1.53832 1.40143 1.242361.07617 0.91018 0.74965 0.58632 0.44389 0.31969 0.21817 0.13628 0.6251.44474 1.31505 1.16949 1.01367 0.85740 0.70376 0.56073 0.42377 0.308930.21079 0.13229 0.650 1.34688 1.22989 1.09583 0.95002 0.80474 0.667670.52945 0.40544 0.29302 0.20195 0.12697 0.675 1.24980 1.14066 1.018860.89847 0.76096 0.62665 0.49851 0.38186 0.27822 0.19282 0.12183 0.7001.15205 1.05395 0.94618 0.82121 0.70183 0.58637 0.46819 0.36001 0.264570.18183 0.11589 0.725 1.06695 0.95985 0.86899 0.76311 0.65554 0.542980.43531 0.33593 0.24719 0.17068 0.10880 0.750 0.95919 0.88016 0.786390.68639 0.59748 0.49300 0.39905 0.30942 0.22837 0.15833 0.10205 0.7750.85579 0.78705 0.70261 0.61881 0.54105 0.45283 0.36509 0.28271 0.210410.14649 0.09400 0.800 0.76510 0.70664 0.62494 0.56082 0.48244 0.405390.32858 0.25609 0.19017 0.13345 0.08550

The maximum velocity (Table 5) did not vary significantly with angle α60, decreasing almost linearly as R₁ 52 increased.

TABLE 5 Angle α (degree) R₁ 40 50 60 70 80 90 100 110 120 130 140 0.2501.59580 1.61260 1.62120 1.62540 1.62740 1.62800 1.62830 1.62820 1.628301.62840 1.62840 0.275 1.52690 1.54460 1.55350 1.55760 1.55960 1.560301.56070 1.56060 1.56070 1.56070 1.56080 0.300 1.45520 1.47380 1.482801.48710 1.48890 1.48950 1.49010 1.48990 1.48990 1.48980 1.49000 0.3251.38187 1.40066 1.40966 1.41406 1.41566 1.41628 1.41686 1.41684 1.416691.41675 1.41685 0.350 1.30762 1.32626 1.33532 1.33909 1.34066 1.341401.34151 1.34175 1.34153 1.34165 1.34175 0.375 1.23271 1.25050 1.258921.26294 1.26438 1.26496 1.26510 1.26528 1.26520 1.26535 1.26507 0.4001.15718 1.17418 1.18224 1.18572 1.18725 1.18776 1.18789 1.18796 1.187841.18802 1.18802 0.425 1.08181 1.09777 1.10522 1.10828 1.10965 1.109941.10998 1.11017 1.11011 1.11028 1.11025 0.450 1.00674 1.02154 1.028161.03098 1.03213 1.03229 1.03263 1.03260 1.03252 1.03260 1.03267 0.4750.93256 0.94562 0.95169 0.95420 0.95519 0.95546 0.95555 0.95551 0.955550.95540 0.95535 0.500 0.85912 0.87085 0.87614 0.87827 0.87908 0.879220.87922 0.87916 0.87931 0.87931 0.87927 0.525 0.78721 0.79767 0.802070.80347 0.80406 0.80425 0.80440 0.80446 0.80453 0.80442 0.80444 0.5500.71674 0.72549 0.72928 0.73066 0.73099 0.73118 0.73114 0.73109 0.731190.73115 0.73098 0.575 0.64826 0.65587 0.65827 0.65957 0.65976 0.659740.65988 0.66004 0.65997 0.65997 0.66000 0.600 0.58185 0.58788 0.590000.59084 0.59125 0.59116 0.59112 0.59098 0.59109 0.59096 0.59109 0.6250.51788 0.52246 0.52416 0.52472 0.52475 0.52492 0.52474 0.52469 0.524900.52485 0.52479 0.650 0.45641 0.45985 0.46121 0.46147 0.46134 0.461600.46145 0.46151 0.46141 0.46159 0.46146 0.675 0.39788 0.40039 0.401110.40144 0.40129 0.40152 0.40150 0.40137 0.40137 0.40141 0.40135 0.7000.34231 0.34415 0.34464 0.34485 0.34486 0.34485 0.34488 0.34471 0.344760.34483 0.34468 0.725 0.29030 0.29143 0.29175 0.29180 0.29187 0.291800.29171 0.29182 0.29186 0.29181 0.29189 0.750 0.24191 0.24260 0.242750.24268 0.24266 0.24290 0.24278 0.24279 0.24275 0.24292 0.24272 0.7750.19733 0.19777 0.19774 0.19779 0.19786 0.19775 0.19777 0.19777 0.197760.19771 0.19779 0.800 0.15689 0.15705 0.15712 0.15710 0.15687 0.156870.15707 0.15704 0.15713 0.15714 0.15702

Example 2 Pressure Gradient Determination

A possible minimal pressure gradient for a cannula as provided hereinwas determined as follows. The pressure gradients for return flowthrough lumens 36 and 37 were calculated for different values of angle α60 and R₁ 52 (Table 6). The minimum pressure gradient occurred atR₂=1.0, R₁≈0.6 to 0.625 and angle α≈90°.

TABLE 6 Angle α R₁ 40 50 60 70 80 90 100 110 120 130 140 0.250 0.32790.3278 0.3278 0.3278 0.3277 0.3277 0.3277 0.3277 0.3277 0.3277 0.32770.275 0.2247 0.2246 0.2246 0.2245 0.2245 0.2245 0.2245 0.2245 0.22450.2245 0.2245 0.300 0.1594 0.1593 0.1593 0.1592 0.1592 0.1592 0.15920.1592 0.1592 0.1592 0.1592 0.325 0.1165 0.1164 0.1164 0.1163 0.11630.1163 0.1163 0.1163 0.1163 0.1163 0.1163 0.350 0.0875 0.0873 0.08730.0872 0.0872 0.0872 0.0872 0.0872 0.0872 0.0872 0.0872 0.375 0.06730.0671 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.06700.400 0.0529 0.0528 0.0527 0.0526 0.0526 0.0526 0.0526 0.0526 0.05250.0525 0.0525 0.425 0.0426 0.0424 0.0423 0.0422 0.0422 0.0421 0.04210.0421 0.0421 0.0421 0.0421 0.450 0.0351 0.0348 0.0347 0.0346 0.03450.0345 0.0345 0.0345 0.0345 0.0345 0.0345 0.475 0.0296 0.0293 0.02910.0290 0.0289 0.0289 0.0289 0.0289 0.0289 0.0289 0.0289 0.500 0.02560.0252 0.0250 0.0249 0.0248 0.0248 0.0247 0.0247 0.0247 0.0247 0.02470.525 0.0228 0.0223 0.0220 0.0219 0.0218 0.0218 0.0217 0.0217 0.02170.0217 0.0217 0.550 0.0210 0.0204 0.0200 0.0198 0.0197 0.0197 0.01960.0196 0.0196 0.0196 0.0196 0.575 0.0199 0.0192 0.0188 0.0185 0.01840.0183 0.0183 0.0183 0.0183 0.0183 0.0183 0.600 0.0197 0.0188 0.01820.0179 0.0178 0.0177 0.0176 0.0176 0.0176 0.0176 0.0176 0.625 0.02020.0191 0.0184 0.0180 0.0178 0.0177 0.0176 0.0176 0.0176 0.0176 0.01760.650 0.0216 0.0202 0.0193 0.0188 0.0185 0.0184 0.0183 0.0183 0.01830.0183 0.0183 0.675 0.0242 0.0223 0.0211 0.0205 0.0201 0.0199 0.01980.0198 0.0198 0.0198 0.0198 0.700 0.0281 0.0256 0.0242 0.0233 0.02280.0225 0.0224 0.0223 0.0223 0.0223 0.0223 0.725 0.0341 0.0307 0.02880.0277 0.0270 0.0266 0.0264 0.0263 0.0263 0.0263 0.0263 0.750 0.04310.0385 0.0359 0.0343 0.0334 0.0329 0.0326 0.0325 0.0324 0.0324 0.03240.775 0.0570 0.0505 0.0468 0.0446 0.0432 0.0425 0.0421 0.0419 0.04180.0418 0.0418 0.800 0.0790 0.0696 0.0642 0.0609 0.0589 0.0578 0.05720.0569 0.0568 0.0568 0.0568

Other Embodiments

It is to be understood that while the invention has been described inconjunction with the detailed description thereof, the foregoingdescription is intended to illustrate and not limit the scope of theinvention, which is defined by the scope of the appended claims. Otheraspects, advantages, and modifications are within the scope of thefollowing claims.

1. A method for implanting a cannula into the heart of a mammal, whereinsaid method comprises: (a) puncturing the heart or blood vessel of saidmammal, (b) inserting a cannula into a chamber of the heart, whereinsaid cannula comprises a housing having a proximal region, a distalregion, and an intermediate region located between said proximal anddistal regions, wherein said housing defines a first lumen and a secondlumen, wherein said first lumen comprises (a) a proximal end locatedwithin said proximal region of said housing and adapted to engage ablood pump and (b) a distal end located within said distal region ofsaid housing and adapted to be positioned within a cardiovascularsystem, wherein said second lumen comprises (a) a proximal end locatedwithin said proximal region of said housing and adapted to engage saidblood pump and (b) a distal end located within said intermediate regionof said housing and adapted to be positioned within said cardiovascularsystem, wherein one of said first and second lumens has a generallyreniform cross-sectional shape in said intermediate region formed atleast by a substantially circular outer arc, a substantially circularinner arc, a first semicircular arc between first ends of the outer andinner arcs, and a second semicircular arc between second ends of theouter and inner arcs, and wherein said distal region is positionedwithin a blood vessel of said mammal and said intermediate region ispositioned within a chamber of the heart of said mammal, and wherein thereniform shape has rounded corners having a radius greater than aboutten percent of the radius of the largest circle that can be inscribedwithin the reniform shape.
 2. The method of claim 1, wherein said methodcomprises connecting a blood pump to said proximal region of saidcannula.
 3. The method of claim 2, wherein said blood pump receivesblood from said heart through said second lumen and pumps blood to saidblood vessel through said first lumen.
 4. The method of claim 3, whereinsaid distal end of said first lumen is positioned in the aorta andwherein said distal end of said second lumen is positioned in the leftventricle.
 5. The method of claim 3, wherein said distal end of saidfirst lumen is positioned in the pulmonary artery and wherein saiddistal end of said second lumen is positioned in the right ventricle. 6.The method of claim 1, wherein each of the semicircular arcs abut thefirst and second ends of the outer and inner arcs.
 7. A cannula for usewith a blood pump, wherein said cannula comprises: a housing having aproximal region, a distal region, and an intermediate region locatedbetween said proximal and distal regions, wherein said housing defines afirst lumen and a second lumen, wherein said first lumen comprises: (a)a proximal end located within said proximal region of said housing andadapted to engage said blood pump, and (b) a distal end located withinsaid distal region of said housing and adapted to be positioned within acardiovascular system, wherein said second lumen comprises: (a) aproximal end located within said proximal region of said housing andadapted to engage said blood pump, and (b) a distal end located withinsaid intermediate region of said housing and adapted to be positionedwithin said cardiovascular system, wherein one of said first and secondlumens has a generally reniform cross-sectional shape in saidintermediate region formed at least by a substantially circular outerarc, a substantially circular inner arc, a first semicircular arcbetween first ends of the outer and inner arcs, and a secondsemicircular arc between second ends of the outer and inner arcs, andwherein the reniform shape has rounded corners having a radius greaterthan about ten percent of the radius of the largest circle that can beinscribed within the reniform shape.
 8. The cannula of claim 7, whereinsaid intermediate region of said housing comprises a single outer wallthat houses said first and second lumen.
 9. The cannula of claim 8,wherein said single outer wall is flexible.
 10. The cannula of claim 7,wherein one of said first or second lumen has a generally circularcross-sectional shape.
 11. The cannula of claim 10, wherein saidintermediate region comprises a septum configured to form a convexsurface of said generally reniform cross-sectional shape and a concavesurface of said generally circular cross-sectional shape.
 12. Thecannula of claim 11, wherein said septum is flexible.
 13. The cannula ofclaim 7, wherein said intermediate region comprises a sensor, sensorleads, or a fluid port.
 14. The cannula of claim 13, wherein said sensoris a pressure transducer, a thermister, an ultrasonic probe, or acombination thereof.
 15. The cannula of claim 7, wherein saidintermediate region comprises a third or a fourth lumen.
 16. The cannulaof claim 15, wherein said third or fourth lumen comprises a fluid port.17. The cannula of claim 7, wherein said housing comprises a singleouter wall in said intermediate region that houses said first and secondlumen, a first outer wall in said proximal region that houses said firstlumen, and a second outer wall in said proximal region that houses saidsecond lumen.
 18. The cannula of claim 7, wherein said distal ends ofsaid first and second lumen bifurcate from said intermediate region ofsaid housing.
 19. The cannula of claim 7, wherein said intermediateregion comprises at least one malleable wire.
 20. The cannula of claim7, wherein each of the semicircular arcs abut the first and second endsof the outer and inner arcs.
 21. A system for providing blood flow to amammal comprising: (i) a cannula, wherein said cannula comprises ahousing having a proximal region, a distal region, and an intermediateregion located between said proximal and distal regions, wherein saidhousing defines a first lumen and a second lumen, wherein said firstlumen comprises (a) a proximal end located within said proximal regionof said housing and adapted to engage a blood pump and (b) a distal endlocated within said distal region of said housing and adapted to bepositioned within a cardiovascular system, wherein said second lumencomprises (a) a proximal end located within said proximal region of saidhousing and adapted to engage said blood pump and (b) a distal endlocated within said intermediate region of said housing and adapted tobe positioned within said cardiovascular system, wherein one of saidfirst and second lumens has a generally reniform cross-sectional shapein said intermediate region formed at least by a substantially circularouter arc, a substantially circular inner arc, a first semicircular arcbetween first ends of the outer and inner arcs, and a secondsemicircular arc between second ends of the outer and inner arcs, andwherein the reniform shape has rounded corners having a radius greaterthan about ten percent of the radius of the largest circle that can beinscribed within the reniform shape; and (ii) a blood pump.
 22. Thesystem of claim 21, wherein said proximal end of said first lumen isconnected to the outflow of said blood pump and wherein said proximalend of said second lumen is connected to the inflow of said blood pump.23. The system of claim 21, wherein each of the semicircular arcs abutthe first and second ends of the outer and inner arcs.